A set of $k$ orthonormal bases of $\mathbb C^d$ is called mutually unbiased if $|\langle e,f\rangle |^2 = 1/d$ whenever $e$ and $f$ are basis vectors in distinct bases. A natural question is for which pairs $(d,k)$ there exist $k$ mutually unbiased bases in dimension $d$. The (well-known) upper bound $k \leq d+1$ is attained when $d$ is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain $C^*$-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of $S_d$ and $S_k$.
We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the $S_d \wr S_k$-module $\mathbb C^{([d]\times [k])^t}$ into irreducible modules. We present numerical results for small $d,k$ and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist $d+2$ mutually unbiased bases in dimensions $d=2,3,4,5,6,7,8$. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than $3$ MUBs in dimension $6$ requires polynomials of total degree at least $12$.

中文翻译：

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互无偏基：多项式优化和对称性

$\mathbb C^d$ 的一组 $k$ 正交基称为互无偏如果 $|\langle e,f\rangle |^2 = 1/d$ 只要 $e$ 和 $f$ 是基向量不同的基础。一个自然的问题是，对于哪些对 $(d,k)$，在 $d$ 维度上存在 $k$ 个相互无偏基。当 $d$ 是素数的幂时，就达到了（众所周知的）上限 $k \leq d+1$。对于所有其他维度，是否能够达到界限是一个悬而未决的问题。 Navascués、Pironio 和 Acín 展示了如何根据某个 $C^*$-代数的存在性重新表述存在性问题。这自然会导致非交换多项式优化问题和相关的半定程序层次结构。该问题具有来自 $S_d$ 和 $S_k$ 的花环积的对称性。
我们利用这种对称性（分析上）来减小半定程序的大小，使它们（在数字上）易于处理。关键一步是将 $S_d \wr S_k$ 模块 $\mathbb C^{([d]\times [k])^t}$ 新颖地显式分解为不可约模块。我们给出了较小的 $d,k$ 和较低层次的数值结果。特别是，我们获得了（众所周知的）事实的平方和证明，即维度 $d=2,3,4,5,6,7,8 中不存在 $d+2$ 相互无偏基$。此外，我们的数值结果表明，在上述框架中，在 $6$ 维度上存在超过 $3$ MUB 的平方和反驳需要总次数至少为 $12$ 的多项式。